I wanted to teach a lesson on using systems of linear equations to solve an interesting problem instead of the typical gym membership problems. I turned to the MTBoS search engine, but struck out (I was being really picky). I did, however, run across Christopher Danielson's "Oreo Manifesto" and found my inspiration. Then I found a picture at residentevents.com that helped inspire the rest of the problem. To spark interest and start the problem solving process, I started the lesson by showing the picture of the oreo stack on my smartboard. I had the students notice and wonder while I wrote what they said on the board. I then shared two things that I wondered: 1) How many calories would this oreo stack be? 2) How many grams would it be? In order to answer these questions, we brainstormed some things we might need to know. We also had a conversation about if the stufs in the oreo stack were from regular oreos or double stuf oreos. We decided to say that they were from regular oreos, but some were convinced that they were from double stuf. I followed up by giving the students the following image. Note: At this point, the students have brainstormed, but I have not told them what they need or how to answer the questions (in reality, I have not told them anything). I used the public brainstorming as a way to let them share ideas, but didn't want to do the thinking for them.
I randomly grouped the students together with cards and sent them to the whiteboards. We decided to work on the first question, "How many calories is the stack?" I let the students play around with the problem for a while. They struggled, which I anticipated. It became the headache and they wanted the aspirin (see Dan Meyer). As they worked, I visited each group to hear their ideas and thoughts. After a few minutes, I called the class together to one of the white boards. I explained that something that helps get a problem going is figuring out what the variables are. "What don't you know that you want to know?" We wanted to know how many calories the stack is. But what do we need to know more specifically? 1) How many calories is a wafer 2) How many calories is a stuf I sent the students back to their boards with the mission to now create equations. Once again, they struggled, but kept trying. I walked around to the groups and visited with them about what they were trying. (When talking with the students, it is my goal to learn what they understand and then move them forward with careful questioning. My rule is that I cannot tell them anything, I must ask a question) After a while, I called the class back together to talk about how to build the equations (aspirin to their headache again). They went back to their boards, wrote their equations and found how many calories were in the stack. (They have already learned how to solve systems). When each group answered the first question, I asked them to work on the second equation, "How many grams is the stack?" They jumped in with enthusiasm and worked until the end of the hour. Some finished, some did not. Either way, we all ate oreos before leaving for the day! To follow up, the next day, we worked on the typical word problems: from the situations, we constructed equations and solved the systems.
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I have always been frustrated with teaching zero exponents and negative exponents. It is difficult to really get the students to understand. But I think I have finally found a good lesson! Of course, it includes visuals that students can experiment with. When we were done with it, I was pleasantly surprised when, in answer to the question, "What is a^0?", both classes answered, "the division by itself." (Which was not a phrase I used in class, but a phrase they created from their exploration).
I struggled at the beginning of the lesson to get them to forget about the exponent rules they memorized and really understand what we were exploring. But, through the process, they forgot about the memorized rules, and embraced the math we were doing. I count this lesson a success! You can get the lesson here. First handout for Stand and Talk Collection of various patterns for groups to work with Last spring, I asked my principal if I could get whiteboards around my room. I didn't have a specific plan, I didn't know any of the research. I had just been thinking about it for a while and felt like I wanted to get my students up and writing on the boards. He agreed and I had boards on my walls a few short weeks later.
I wasn't sure when I would get a chance to try them out, but I got inspiration the first day and had my students at the boards. I cannot even remember what I had them doing, but I remember watching them working and being engaged in the math and I was hooked. I had them do problems on the boards a couple days a week. Sometimes it would be homework problems, but I also had them doing open middle problems and some problem solving. I finished the year pleased with how the boards were going, but displeased with how little effort my students were willing to put into application problems. We wrapped up the MCAs (MN state test), and although my students did well, I kept thinking that if I could only get my students to THINK, they would do so much better. They knew their math, they just weren't making the effort to apply it to the application problems. Fast forward to summer. I like to listen to math education podcasts as I go for walks. (Don't tell my friends, they will think I am nerdier than they already know I am). I happened upon the Global Math Department webinar recording with Peter Liljedahl. You can find it here. I was inspired! It was exactly what I was looking for. It was called the Thinking Classroom. The Thinking Classroom starts with three main steps: 1) A good problem (or series of problems) 2) Visibly Random Groups 3) Vertical Non-Permanent Surfaces (I just got those!) I went back to school in the fall and was ready for the change, sort of. I at least had a plan to incorporate the Thinking Classroom. I didn't exactly have the lessons figured out, but I had some inserted here and there. I did have the students at the boards a lot, but it wasn't exactly the picture-perfect Thinking Classroom. I had a day-long workshop that I had to present at. I decided that the Thinking Classroom would be a great topic. So I started researching and experimenting. Here is the cool thing about experimenting with this, it was so easy. I didn't have any days where I wondered if this was a good idea. Why? Because, when students are thinking it is a great day! When I first learned about the Thinking Classroom, I thought I would always have to do these great problems like what Peter shares on his website. But, what I found is that I could also take mundane, procedural problems and turn them into a great lesson at the boards. I became hooked on the teaching structure. My students are at the boards almost every day, at least for part of the day. However, one thing I didn't find anywhere in my research until I asked the question on Twitter: there are still plenty of days where the students are working individually at their desks. You are not expected to spend every minute at the boards, although there are some that do. You need to make it work for you. I am sure I will grow and change as I continue to teach this way. But for now, I still have homework assignments and follow their board time with flipped videos to help tie everything together. I recently gave an hour presentation at our MCTM conference. The room was overflowing. Teachers are hungry to find a way to get their students to think. I highly recommend everyone tries to do at least some form of this. Here is my presentation, the second slide has the recording of the hour. The last slide has lots of resources for you to learn more. Feel free to contact me if you have questions. P.S. I am working on creating a place to store my lessons to share with others. I will hopefully have a good collection posted on my website by the end of the summer. There are some examples of my lessons in my presentation above. In Minnesota, our state test is called the MCA. Unfortunately, it ends up a focus for us math teachers, especially in the high stakes 11th grade. I am not different. I have spent way too much energy focused on what I can do to get my students to perform better on the test. I teach in a small school district, EVERY 11th grader is my student. So the scores of my district's 11th grade reflects on my class.
As we prep in the month before the test, I often find myself frustrated that my students know how to do the math, but look at the problem in front of them as if they have never seen it before. I encourage them by saying, "You have done this, you know how to do this!" And yet, they end up guessing at an answer. So then we spend more time talking about it and I spend my summer trying to figure out how I can teach it better. But I had an epiphany last spring, I have not been teaching my students to think, I have been teaching them to mimic me. Scratch that, I have not been letting my students think, I have been sucking the thinking out of my classroom by over- scaffolding and over-helping and over-lecturing! I have robbed them of the chance to think by doing the thinking for them, and then getting frustrated when they don't think for themselves. I have had the best intentions and done the best that I knew how, but it was me that stopped them from thinking. So now what?!?!?! How do I teach them to think? I have tried giving students problems to solve on their own or in groups. It has not worked the best. My first start was trying stand and talks that I had learned about from Sara Vanderwerf. (Webinar)They made such a difference in my class. I am constantly asking myself, "What can I get the kids to say instead of telling them?" I have created images, borrowed from others, and spontaneously used work from class. My students have been way more involved in the curriculum and retain it so much better. Every teacher, at every level, NEEDS to implement this routine daily. Side note: I have been bragging up this routine for quite a while now, but I am not sure people really understand it by my description. The other day, my principal came into my classroom to ask me a quick question. I was just handing out my image for a stand and talk. I finished and visited with him for 30 seconds. I guess he was intrigued as to why my students were standing around the room during class, so he stood there and watched. It was fun to watch the expression on his face as my students visited about the image with their partner and then shared out with the class. He commented later about how impressed he was with how involved everyone was and that even the ones who typically don't engage in classes had something to notice. Next, I was lucky enough to run across a couple webinars that changed my classroom forever: Building Thinking Classrooms and What Thinking Classrooms Look and Feel LIke. They made so much sense and fell right into what I wanted to change in my classroom! So I did what I always do, jumped in with both feet! I tried two different things this year to celebrate Pi Day in my classroom. My high school students have already done the string to measure the circumference and diameter in previous years. I wanted to do something more active than watching youtube videos. So I watched for ideas on twitter. (If you are not a twitter user, please consider getting an account and trying it!) The first activity I found was Pi Poems. This came from the great blog by Eric Curts. He has so many great technology ideas for various subject areas. (Here is the Pi Poem blog post). The structure of a Pi Poem is that the number of letters in the word match the corresponding digit in Pi. The first word is 3-letters long, the second is 1-letter long, the third is 4 letters long and so on. I teamed up with the ELA department for this assignment. We gave them some guidelines like: the poem had to be Pi-themed. Basically, it had to be about pi or something circular. They also had to create a final product that was creative. (We are going to display their poems at our spring Art Show!) Here is the google doc we used to give them their guidelines. We also supplied the students with Eric Curts' spreadsheet to help keep track letters in the words. I loved the resulting poems. They were unique and clever. See below for some of my favorites! The second activity I did was a twitter project I found on Alice Keelers blog. The original activity was designed by Ashley Fort for historical figures as the subject. I tweaked it to be about pi. My students were told to research pi, its history, and its uses. On Pi Day, I gave them the template and final instructions to make a mock twitter page for pi. They had to come up with a handle, the bio, a picture, and then tweet as pi and also as someone tweeting to pi. I enjoyed what they came up with, however, we had a little problem with just copy and pasting what they found on websites. When they shared their work with the rest of the class, we had a good discussion about plagiarism and how to appropriately share someone else's words. Here is the result of that day. Some of the Pi Poems from March 14th, 2018
It is the age old frustration for anyone teaching math. How do we get kids to be better at solving word problems? There are many answers out there; many opinions. But let's be honest, many of us are hoping there is an easy answer that won't require a new curriculum, hours of prep and completely revamping our teaching methods. Although I think that all three things are often needed, it is not reality. About a year ago, I discovered I had effectively isolated my professional life to within the four walls of my classroom since 2009. I think it was a combination of raising children and wearing too many hats within my school-district. Luckily, I broke free! And one of the first things I learned was about the simple, but impactful, question "What do you Notice?". Really Quick Rant: How is this not the cornerstone of preservice teacher's education? In case you are like I was a year ago, "What do you Notice?" is a question you ask students whenever you want them to notice things, expand their thinking, or take time before they dive into a question. Basically, don't notice/say anything that the kids can notice/say. If you haven't seen it, watch Annie Fetter's ignite video
Applying this to word problems has completely changed my classroom.
Whenever we are going to work on a word problem, I REMOVE THE QUESTION. How many times have you asked or been asked, "How do I get kids to spend more time on word problems before just 'doing' the math they think it wants?"? It is simple, remove the question. Next: Ask the kids what they notice. When you are first doing this, it is important to MODEL, MODEL, MODEL. Have the students popcorn out what they notice and write it all on the board. And I mean ALL. Give each and every thing that is noticed validation by writing it on the board. You are letting EVERY student know that what they notice is valuable and encouraging them to keep noticing. When you have exhausted everything that can be noticed, ask the kids to come up with the question that can be asked and then solve it. Because they just spent so much time noticing, they are typically ready to do the math without the support they would have required in the past. You can/should extend this to be done individually also. Take a piece of paper and put the problem in the middle with a circle around it (no question again). Have the students write everything they can notice around the circle in a web-format (concept map). Then at the bottom ask them to come up with a question and solve it. What can you expect:
In reflection, it is embarrassing that this was not an intuitive thing to do with my students. Instead, I was always reading the problem outloud, then jumping to the question and asking the students what information they need. I would then try to get them to dissect the information, but they were too eager to just do math with the numbers given and get the problem done. By establishing this model, I am hoping students will be more apt to do this on their own. And I am hoping to see a change in how they attack their state test in the spring. Goals: 1) Students understand what they are doing when they solve equations, beyond memorized steps 2) Students see/understand the efficiency of getting rid of the constants first 3) Students know they can multiply/divide first, but also understand that they have to do it to the entire side, not just the term with the variable 4) Students continue to keep their equations balanced 5) Students continue to make connections between solving equations abstractly and solving with Algeblocks Lesson: Day 1: I began this lesson with a number talk. The idea is to give them one that they have to do in a couple steps and then relate the lesson back to it at the end of the hour. Number Talk: (See picture above) I have six tables. I told the kids that I had 20 pencils that I wanted to put in the resource boxes on the tables. If I put the same number of pencils in each box, how many did I put in each box? The students shared about 5 different solutions and we went on to the lesson. (I wish I had taken a picture of these. I thought I did, but cannot find it any where on my phone). Next I wanted the students to explore solving a two-step equation. I decided to use the balance scales because I wanted the students to see it being balanced or unbalanced after their choice of operation. (We have been talking a lot about how the equal sign in an equation means that it is balanced). The equation I used was 2x+3=9. My paraprofessional and I set up the balance scales at each table. Before we started solving, I asked the students what they noticed about the scales. This step is something I was missing in the past. It is important to take the time to make sure students have studied what is in front of them before asking them to make decisions on how to solve it. They noticed: "There are two canisters" (We wondered and confirmed that they were the same weight and worth) "There are 3 white tokens with the canisters" "There are 9 white tokens on the other side" "The two sides are balanced, so they weigh the same" "The equation is 2x+3=9" At this point I told the students that I wanted them to figure out how much ONE canister weighs, but it is important that they "have a math reason for what they are doing". (There are probably better ways of saying this, but my ninth graders nodded their heads and got busy, so it must have made sense to them). There was great discussion within the groups on how to solve the equation. They talked about making sure they did the same thing on both sides and that they needed to keep it balanced. They pretty quickly decided to remove the three tokens from each side. Dividing by 2 was a bit trickier with the balance scales, for two reasons. First, when we worked with Algeblocks to do this with one-step equations, you could easily split the pieces into two groups. But the pans of the balance scale are not big enough to do this. So the students had to remove one of the two groups. Second, because of the actual balancing of the scales, they could "cheat" to figure out how many tokens to leave on the right. They would just remove one token at a time until it was balanced, instead of splitting the tokens in half and removing one of those groups. At this point I had to have them put the pieces back on and ask them to talk me through their steps. To wrap up this problem, I had the students set up their scales again with the same equation and I wrote the equation on the board. As a whole class, they talked me through their steps and then we discussed how we could write the same thing with the equation. See our work below. I next wanted to make a point about the efficiency of removing the constants first, but also point out that you could divide/multiply first (as long as you do it correctly). I chose to have the students solve the equation 2x+4=10 on the balance scales. I wrote the equation on the board and had them set it up on their scales. Note: I made sure the value of x stayed the same so that they could just use the canisters as they were. I also chose even numbers so that they could physically divide them all by 2. My instructions to the students were: "I want you all to solve this equation on the scales. When you are done, I want you to put it back on and solve it a different way the second time. Make sure what you do to one side, you do the exact same thing to the other and make sure you keep it balanced". The students quickly solved it the first way by removing the constants (extra tokens) first and then dividing into two groups, just like the first problem. When they went to solve it a second way, one of three things happened: 1) They didn't know what to do and sat somewhat quietly 2) They tried different things and maybe ended up at the answer but with sketchy math steps (see previous trick from first problem) 3) They did well with dividing both sides by 2 first and figuring out that you also have to divide the four tokens. Once again, I had all the groups put the equation back on their scales. As a class, we walked through the first solution while they did the scales and I did the algebra on the board. Then I had them put the equation back on the scales so that we could talk about the second step. Knowing that very few figured it out, I decided to try to lead the groups to the second method. One of my favorite questions to ask during these solving equations lessons is "What is our goal?". I find students need to be reminded that they are trying to solve for x, which means they are trying to get x by itself. So I asked that and then followed with, "What are the two things we need to get rid of?" Students knew they wanted to get rid of the 2 and the 4. They also realized at this point, that if they got rid of the 4 first last time, they will want to get rid of the 2. To purposely mislead them, I showed dividing the 2x by 2 and the 10 by 2 on the board (I did NOT divide the 4 by 2). I wanted them to see the unbalanced result on the balance scale. It was interesting to see the reactions of students. Some eagerly went to the scales to do it, but there were a few that started to protest. I quietly nodded in agreement to them and they quickly accepted my choice (I am thankful that they have gotten to know me well enough by now to trust my teaching moves). The students pretty quickly realized the mistake of not dividing the four. It was great to hear their conversations of why they were not balanced and how to fix it. We reconvened on the board and discussed as a class what happened. They did a great job justifying why we need to divide EVERYTHING, not just the 2x and 10. You can see our work below. But there was an important discussion to have still. Which method should we use? When should we use each method? Is there a more efficient method? So, we went back to our first problem, 2x+3=9. I had the students set it back up and then told them to solve it by dividing first. I barely started to walk around the room before the students started protesting. I challenged them to keep working it out, but it didn't last long before many were giving up. We gat |